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Counting triangles with same centroids

Source: RMO Maharashtra and Goa 2016, P6

October 11, 2016

countingMass Pointsgeometry

Problem Statement

ABC

is an equilateral triangle with side length

11

units. Consider the points

P_1,P_2, \dots, P_10

dividing segment

BC

into

11

parts of unit length. Similarly, define

Q_1, Q_2, \dots, Q_10

for the side

CA

and

R_1,R_2,\dots, R_10

for the side

AB

. Find the number of triples

(i,j,k)

with

i,j,k \in \{1,2,\dots,10\}

such that the centroids of triangles

ABC

and

P_iQ_jR_k

coincide.

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